Question:
Consider the intervals \( S = (0, 2] \) and \( T = [1, 3] \). Let \( S^\circ \) and \( T^\circ \) be the sets of interior points of \( S \) and \( T \), respectively. Then the set of interior points of \( S \setminus T \) is equal to
Consider the intervals \( S = (0, 2] \) and \( T = [1, 3] \). Let \( S^\circ \) and \( T^\circ \) be the sets of interior points of \( S \) and \( T \), respectively. Then the set of interior points of \( S \setminus T \) is equal to
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The set of interior points of \( A \setminus B \) is \( A^\circ \setminus B^\circ \), where \( A^\circ \) and \( B^\circ \) are the interior points of \( A \) and \( B \), respectively.
Updated On: Dec 11, 2025
- \( S \setminus T^\circ \)
- \( S \setminus T \)
- \( S^\circ \setminus T^\circ \)
- \( S^\circ \setminus T \)
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The Correct Option is B, D
Solution and Explanation
Step 1: Define the interior points.
The interior points of a set \( A \) are the points of \( A \) that have a neighborhood entirely contained in \( A \).
Step 2: Find the interior points of \( S \setminus T \).
The interior points of \( S \setminus T \) are the points that belong to \( S^\circ \) (the interior of \( S \)) but do not belong to \( T^\circ \) (the interior of \( T \)). Hence, the set of interior points of \( S \setminus T \) is \( S^\circ \setminus T^\circ \).
Step 3: Conclusion.
Thus, the correct answer is \( \boxed{(C)} \).
The interior points of a set \( A \) are the points of \( A \) that have a neighborhood entirely contained in \( A \).
Step 2: Find the interior points of \( S \setminus T \).
The interior points of \( S \setminus T \) are the points that belong to \( S^\circ \) (the interior of \( S \)) but do not belong to \( T^\circ \) (the interior of \( T \)). Hence, the set of interior points of \( S \setminus T \) is \( S^\circ \setminus T^\circ \).
Step 3: Conclusion.
Thus, the correct answer is \( \boxed{(C)} \).
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